2 BELL SYSTEM TECHNICAL JOURNAL 



It has been stated by the earlier authors that the curvature of the wave 

 guide produces a coupling between modes. Before going into a detailed 

 analysis one may estimate by inspection the nature of this coupling and the 

 kind of modes that are most strongly coupled to each other. Figure la 

 shows the cross section and the longitudinal section of a straight cylindrical 

 wave guide. The location of every point inside the wave guide is determined 

 by three coordinates: the radial distance r from the cylinder axis; the 

 azimuth angle (p from an arbitrary line and the axial distance z from the 



a- CYLINDRICAL CO-ORDINATES IN STRAIGHT WAVEGUIDE 



TOROIDAL CO-ORDINATES IN CURVED WAVEGUIDE 



Fig. 1 



origin. If the wave guide is bent as shown on Fig. lb, but a wave front 

 at right angles to the cylinder axis is to be maintained, the waves must be 

 shortened at the inside of the bend and lengthened at the outside of the 

 bend. Regarding compression as a positive and expansion as a negative 

 deformation, one sees that the distortion of the wave shape is proportional 

 to the curvature of the wave guide multiplied by the cosine of the azimuth 

 angle. It is natural to assume that the coupling between modes is propor- 

 tional to this distortion. 



Now it is known that all modes of propagation in a circular wave guide 



